Notes on monotonically metacompact generalized ordered spaces

In this paper, we show that any generalized ordered space X is monotonically (countably) metacompact if and only if the subspace X - { x } is monotonically (countably) metacompact for every point x of X and monotone (countable) metacompact property is hereditary with respect to convex (open) subsets in generalized ordered spaces. In addition, we show the equivalence of two questions posed by H.R. Bennett, K.P. Hart and D.J. Lutzer.

The partially pre-ordered set of compactifications of Cp(X, Y)

In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:(a) Y is not locally compact,(b) X has only one non isolated point and Y is not compact.Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:(i) X has a non-isolated point with countable character,(ii) X is not pseudocompact,(iii) X is infinite, pseudocompact and Cp(X) is normal,(iv) X is an infinite generalized ordered space.Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

A NOTE ON PARACOMPACT p-SPACES AND THE MONOTONE D-PROPERTY

For any generalized ordered space X with the underlying linearly ordered topological space Xu, let X* be the minimal closed linearly ordered extension of X and $\tilde {X}$ be the minimal dense linearly ordered extension of X. The following results are obtained. (1)The projection mapping π:X*→X, π(〈x,i〉)=x, is closed.(2)The projection mapping $\phi : \tilde {X} \rightarrow X_u$, ϕ(〈x,i〉)=x, is closed.(3)X* is a monotone D-space if and only if X is a monotone D-space.(4)$\tilde {X}$ is a monotone D-space if and only if Xu is a monotone D-space.(5)For the Michael line M, $\tilde {M}$ is a paracompact p-space, but not continuously Urysohn.

Dense Sδ-diagonals and linearly ordered extensions

<p>The notion of the S_δ-diagonal was introduced by H. R. Bennett to study the quasi-developability of linearly ordered spaces. In an earlier paper, we obtained a characterization of topological spaces with an S_δ-diagonal and we showed that the S_δ-diagonal property is stronger than the quasi-G_δ-diagonal -diagonal property. In this paper, we define a dense S_δ-diagonal of a space and show that two linearly ordered extensions of a generalized ordered space X have dense S_δ-diagonals if the sets of right and left looking points are countable.</p>